CAREER: Flexible and Efficient Exploration of the Bayesian Framework for High Dimensional Modeling

现代大数据时代给统计学家带来了独特的机遇，也带来了挑战。虽然大数据革命带来了从丰富的数据中获得有价值和深刻的见解并增强数据驱动的决策的巨大机会，但它也带来了统计学家和数据科学家在三个关键方面对创新和知识发现的挑战性要求：(I)开发能够适当描述数据复杂性的灵活模型；(Ii)高效有效的统计估计和推理程序；以及(Iii)开发可扩展到大数据集的计算算法。这个项目的目的是通过充分探索贝叶斯框架在这三个方面都取得进展，该框架将模型的参数视为随机的，并提供了一种有效的机制来量化模型参数的不确定性。特别是，开发的技术将有助于分析包含大量协变量的数据集，学习大量结果变量之间的相关性结构，并通过在不同分位数水平上对协变量的关系进行建模来获得对结果变量的影响的全面描述。这项研究将对不同学科的统计实践产生影响，包括生物学、经济学、环境科学、营销学和医学。培训部分将把研究融入教学，在拟议研究的基础上向研究生提供专题课程，并制定在可接触的一级纳入研究概念的本科生研究项目。国际统计研究所将指导高中研究项目，并组织一次K-12外联讲习班，向高中生和教师介绍现代统计及其应用。统计严谨和计算高效的贝叶斯方法和推理程序将被开发，它将适用于各种复杂的高维模型，包括广义线性模型、分位数回归模型和图形模型。提出了贝叶斯正则化先验的一般类，并针对各种常用的似然函数，严格研究了它们的正则化性质。与大多数现有的侧重于高维估计的贝叶斯方法不同，将开发一种新的贝叶斯框架，用于执行具有有效频率特性的高维贝叶斯推理。将设计可扩展的计算技术，该技术不涉及从后验获得点估计以及对全部后验分布进行采样的大型矩阵运算，并将研究其统计特性。计算发展的一个吸引人的特点是，它们将适用于实践中常用的各种统计模型。这项研究将与现代统计学的几个高度活跃的领域密切相关，包括高维建模、贝叶斯计算、非凸正则化、选择后推理、图形模型和分位数回归，并将有助于这些领域的进步和相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The modern era of Big Data brings unique opportunities as well as challenges to the statistician. While the Big Data revolution brings a great opportunity to obtain valuable and profound insights from the richness of data and to enhance data-driven decision making, it also brings challenging demands for innovation and knowledge discovery in three crucial aspects from statisticians and data scientists: (i) development of flexible models that can appropriately describe the complexities of the data (ii) efficient and valid statistical estimation and inferential procedures, and (iii) development of computational algorithms that scale-up to large datasets. The purpose of this project is to make advances in all the three aspects by fully exploring the Bayesian framework, which treats the parameters of a model to be random and provides an efficient mechanism to quantify the uncertainty of the model parameters. In particular, the techniques developed will be useful for analyzing datasets containing a large number of covariates, for learning the dependence structures between a large number of outcome variables, and for obtaining a comprehensive description of the impact of covariates on outcome variables by modeling their relationships at different quantile levels. The research developed will have impact on statistical practice in various disciplines including biology, economics, environmental sciences, marketing, and medical sciences. The training component will integrate research into teaching by offering special topics courses to graduate students based on the proposed research and by developing undergraduate research projects that incorporate research concepts at an accessible level. The PI will mentor high school research projects and organize a K-12 outreach workshop to provide exposure to modern statistics and its applications to high school students and teachers. Statistically rigorous and computationally efficient Bayesian methodologies and inferential procedures will be developed which will be applicable for a variety of complex high dimensional models including generalized linear models, quantile regression models, and graphical models. General classes of Bayesian regularization priors will be proposed, and their regularization properties will be rigorously studied for a variety of commonly used likelihood functions. In contrast to most of the existing Bayesian approaches that focus on high dimensional estimation, a novel Bayesian framework for performing high dimensional Bayesian inference having valid frequentist properties will be developed. Scalable computational techniques that do not involve large matrix operations for obtaining point estimators from the posteriors as well as for sampling the full posterior distributions will be devised and their statistical properties will be studied. An attractive feature of the computational developments will be that they will be applicable to a diverse range of statistical models commonly used in practice. The research developed will be closely related to several highly active areas of modern statistics including high dimensional modeling, Bayesian computation, nonconvex regularization, post-selection inference, graphical models, and quantile regression, and will contribute to the advancement of and interaction between these areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.